How to solve equations of higher degrees

The solution to most equations of higher degrees does not have clear formulas such as finding the roots of a quadratic equation. However, there are several ways to bring that allow you to convert the equation of higher degree to a more clear mind.

The most common method of solving equations of higher degrees is factorization. This approach is a combination of the selection of the integer roots of the divisors of the free term, and the subsequent division of the total polynomial by binomials of the form (x – x0).

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For example, solve the equation x^4 + x3 + 2·x2 – x – 3 = 0.Solution.A free member of this polynomial is -3, therefore, its integer divisors can be numbers ±1 and ±3. Substitute them into the equation and see if it will work identity:1: 1 + 1 + 2 – 1 – 3 = 0.

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So, the first estimated root gave the correct result. Divide the polynomial equation (x - 1). Polynomial division is performed in a column differs from a conventional division only in the presence of the variable.

The discriminant is negative, then real roots the equation has no more. Find the complex roots of the equation:x = (-2 + i·√11)/2 and x = (-2 – i·√11)/2.

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Write down the answer:x1,2 = ±1; x3,4 = -1/2 ± i·√11/2.

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Another method for solving equations of higher degree – change of variables to bring it to the square. This approach is used when all of the degree of the equation is even, for example:x^4 – 13·x2 + 36 = 0